In early May, Sears Holdings (NASDAQ:SHLD) announced that it would begin working with Amazon.com (NASDAQ:AMZN) in the tire business. Under the latest partnership between Sears and Amazon, customers buying tires online from the e-commerce giant can choose to have them shipped to a nearby Sears Auto Center for installation. Appointment scheduling is integrated into the Amazon checkout process.

Initially, only 47 Sears Auto Centers participated in this Amazon tire-installation program. But earlier this week, Sears announced that the program has expanded nationwide to all Sears Auto Center locations. This news caused Sears Holdings stock to spike on Tuesday.

The exterior of a Sears full-line store

Image source: Sears Holdings.

However, these gains probably won't last long. Partnering with Amazon on tire installations is a sensible move for Sears, but it comes way too late to save the company from a death spiral.

The tire partnership goes national

When Sears Holdings first announced the tire-installation program for Amazon customers back in May, it said that the plan was to quickly roll out the service nationwide. A month later, it had already added another 71 Sears Auto Centers to the program, more than doubling the number of participating locations. This was two months ahead of schedule, according to the company.

Thus, the fact that Amazon customers can now have their tire purchases shipped to any Sears Auto Center couldn't have been much of a surprise to investors.

Management's positive commentary about the partnership is a more likely reason for the stock's pop. A blog post on the Sears Holdings corporate site noted that the San Bruno, California store is leading the way with the most Amazon orders so far: at least four to six a day. Most of the customers are new to Sears, which is good news for the company's customer-acquisition efforts.

On the other hand, getting a few new customers a day is a drop in the bucket relative to the scope of Sears Holdings' woes. That's why the stock is still down more than 60% since early May, when Sears first announced its tire partnership with Amazon.

SHLD Chart

Sears Holdings Year-to-Date Stock Performance. Data by YCharts.

The number of auto centers is shrinking -- rapidly

Another reason for skepticism is that Sears is closing auto centers along with its full-line stores at a rapid pace. Sears Holdings operated 445 auto centers as of the beginning of fiscal 2018 -- including three under its new DieHard Auto Center brand -- mostly co-located with Sears stores. That was down from 617 just a year earlier.

The carnage has continued. Sears Holdings has already closed or announced the closing of more than 30% of its remaining full-line Sears stores since the beginning of fiscal 2018. Typically, the auto centers close when the full-line stores go out of business. As a result, by year-end, the number of remaining Sears Auto Centers could be close to 300.

It's hard to get excited about the growth potential of the Amazon tire partnership when Sears is closing stores (and auto centers) left and right.

Even if Sears survives, there will be plenty of additional auto-center closures in the next few years. Dozens of auto centers are located in buildings now owned by Sears Holdings real estate spinoff Seritage Growth Properties, which has the right to evict Sears in order to renovate those auto-center buildings for higher-paying tenants. That includes the San Bruno, California auto center that has apparently seen the best results from the Amazon partnership.

Sears Holdings has entered similar sale-leaseback agreements with other real estate owners in recent years. This paves the way for further forced downsizing of the Sears Auto Center chain.

At best, this makes the auto-center business salable

Sears Holdings operates 19 freestanding auto centers. In theory, traffic from the Amazon tire-installation partnership could support expansion of Sears' freestanding auto-center concepts.

However, this is never going to be a very lucrative business due to cutthroat competition. The best-case scenario is that a modestly profitable chain of freestanding auto centers could be sold or spun off to raise cash needed to plug the holes in Sears Holdings' leaky balance sheet.

Yet even this may be unrealistic. There's nothing to stop Amazon from cutting deals with other (much larger) chains of auto-care centers in the future. Sears' lack of capital to invest may prevent it from capitalizing on this opportunity in any way.

John Mackey, CEO of Whole Foods Market, an Amazon subsidiary, is a member of The Motley Fool's board of directors. Adam Levine-Weinberg owns shares of Seritage Growth Properties (Class A). The Motley Fool owns shares of and recommends Amazon. The Motley Fool has a disclosure policy.