This will be a relatively quick and dirty skim over history, but I hope it does something to dispel this belief. The comment you're replying to ignores a number of salient factors. It assumes that natural market activity resulted in Standard Oil's declining market share. It certainly was a relevant factor, but the overarching history of Standard Oil will hopefully lead you to recognize that it did not act alone.
The Sherman Antitrust Act was passed in 1890 but it was a toothless tiger until Roosevelt assumed the office of president in 1901. Prior to then, the main case involving the act was successful in using it against labor unions. Following 1901, litigation broke up a number of companies. This is important.
Investigation at the time indicated that Standard Oil employed predatory pricing up until 1900. Following 1900, they resorted to a shell-game to obfuscate what was going on.
Amongst the first wave of anti-trust litigation targets were J.P. Morgan's holdings. He was cornering the railroad and steel industries successfully, the biggest coup being the formation of US Steel in 1901. The other crown jewel in the strategy was oil, namely Standard Oil, which operated synergistically with J.P. Morgan's holdings in rail and steel. During this time, we had characters like Elbert H. Gary, CEO of US Steel, who held cross-industry dinners aimed at setting uniform pricing for steel and providing management across businesses with consolidated positions against labour organization. While it had been growing at a steady clip beforehand, Standard Oil's value began to skyrocket as of 1901 with no real change in its core business practices.
The SCOTUS' exposition of the issue does a great job showing how Oil's monopoly was mediated in large part through Rail (the case also indicates that practices wrt pipelines were analogous to rail):
"the bill alleged that the combination and its members obtained large preferential rates and rebates in many and devious ways over their competitors from various railroad companies, and that by means of the advantage thus obtained many, if not virtually all, competitors were forced either to become members of the combination or were driven out of business"
Can't transport oil if you don't have rail cars. Can't build rail cars or rail if you can't make steel, and you can't run your trains or your steel plants without oil.
In 1902, the Roosevelt directed the AG to commence litigation to break-up of the J.P. Morgan headed railroad trust - Northern Securities Company. This matter was such a high priority that the case was resolved and reported through all levels of appeal in 1904. As a side note, this is mind-bendingly fast for competition litigation, even today.
With the threat of anti-trust litigation preventing outright predatory pricing and "competitive" rail back on the table steel and oil began to lose their moats. But "competitive" rail wasn't very competitive, and continued a legacy of providing obfuscated sweetheart deals to Standard. They got broken up 7 years later as a result. By the time steel's litigation hit in 1920, it had proper competition and did not get broken up.