The opposite might be true and that its possible the MLSE BOD has Babcock's back.
Keep in mind Babcock salary is guaranteed and also paid by ownership not management of which Dubas resides.
Babcock's contract;
[On May 20, 2015, it was announced that Babcock would become the new head coach of the Toronto Maple Leafs. He reportedly signed an eight-year contract worth $50 million (an average of $6.25 million per season), and will be the highest-paid coach in NHL history].
So that means Babs is only about 1/2 way through his 8 year deal which means ~$25 mil balance remaining.. Which means MLSE would need to be fully onboard financially with handing Babcock a cheque for $25 mil to go away, and then have to overlap paying the next coach likely Sheldon Keefe also. Better be a pretty good case for setting $25 mil on fire to change the current situation.
Babcock is also a proven coach that delivers the team to the playoffs annually with a long record of success, including leading Canada's teams internationally, while Keefe is a young unproven NHL coach with Zero coaching experience at the NHL level.
So how would the Leafs Owners know that move will increase team success or worsen their chances of making the playoffs annually? They better be prepared to answer to their shareholders and investors that this will increase team success and profitability, and not cause missing the playoffs and tossing away playoff revenue and changes at the Cup.
PS. Its easy for fans to say Fire Babacock as its not their money $$, but for MLSE its a business and so this move of a coaching change would carry a very large payout settlement, so it better be made with that in mind.