Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property

(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.

McCoy showed that, if $R$ is an algebraically closed field, then $A,B$ are simultaneously triangularizable (noted ST). Else, McCoy, again, gave a condition that is equivalent to (*) in

A theorem on matrices over a commutative ring, Bulletin of AMS, 45 (1939) 740-744, in free access here: http://www.ams.org/journals/bull/1939-45-10/S0002-9904-1939-07070-5/S0002-9904-1939-07070-5.pdf

Unfortunately, the previous condition seems (to me) almost useless.

Then my question is: if the matrices, over a ring, $A,B$ satisfy (*) and are separately triangularizable, then are they ST ?

Thanks in advance.