A high stock price can make the company look prestigious. E.g. A company trading at $1,000 per share will be perceived as more valuable even though the firm's market capitalization may be the same as a company whose shares trade at $50.
Share values mustn't fall too low! There have been cases where companies have split shares only to see the stock market dive, pushing shares below $10. Psychologically, this may turn off some shareholders, and in extreme cases, share prices may be too low for a company to be listed on an exchange. Companies will avoid splitting to protect themselves from this possibility. Doubtless large caps wouldn't fancy themselves dropping to being a penny stock!
Why don't they reverse stock split to uplift their share prices?
Stock splits (and reverse-splits) are a zero-sum activity (minus what costs there are to actually implement the split). It's like trading 10 dimes for a dollar bill. The company is not worth any more or less after the fact. It used to be that a lower share price was desired since it would enable more investors that had to buy shares in 100-share lots. Nowadays, with index funds dominating the market, that benefit has largely gone away, which is why you see companies like Amazon and Apple decide not to split their shares that are worth hundreds or thousands of dollars per share.
A reverse split may be required in order to stay listed on exchanges, but it is often seen as a large negative signal, much like suspending a dividend.
So companies have no financial incentive to split or reverse split, but often have behavioral incentives not to reverse split.
My answer is: for what reason?
From your link: "Finance professors have examined stock splits and see no actual impact on a company’s value or performance."
It would follow that reverse stock splits are the same.
It does make sense to keep the share price above the minimum required for listing. In addition, it may make sense to keep the price within certain bounds to ensure a particular lot size.
The psychological argument is a bit dubious. If the share price goes up, that may inspire some (misinformed) people to sell, with the likelihood of driving the price down. If anything, splitting is more likely to provide a better psychological outcome as people don't want to part with shares when they are very low, and more inclined to buy them when they are at "bargain" prices.
Maybe you are thinking of a buy-back?
It is another way to reduce the number of issued shares.
If a company has enough cash on hand, the company buys its shares back (but only from those who are wanting to sell).
If the share prices had fallen much below the book value, the company can actually increase the value of the remaining shares (and thus potentially the price)
Just by being a willing buyer it can prevent the share price from falling so far below the book value that it would be attractive to asset strippers.
The book value is pretty much the net asset value divided by the number of shares