No. They have already allowed Starlink competitors to launch on Space X rockets.
There are multiple players working on constellations of low-orbit satellites competing with Starlink
There are multiple players working on constellations of low-orbit satellites competing with Starlink
Starlink "pays" for launches at cost. While we don't know what SpaceX's cost margins are, they are not trivial. To setup a low orbit constellation is extremely expensive and competitors lose millions per launch that Starlink gets to reinvest.
There's been 136 launches of Falcon 9 for Starlink. ~US$62m per launch? If their margins are 20% that's that's $1.6b in savings. And I bet F9's margins are closer to 50% - supporting Starship and more.
Starlink also has launch priority. Good luck with getting 50 launches a year as a customer...