Strongly clean triangular matrix rings with endomorphisms

‎A ring $R$ is strongly clean provided that every element‎ ‎in $R$ is the sum of an idempotent and a unit that commutate‎. ‎Let‎ ‎$T_n(R,\sigma)$ be the skew triangular matrix ring over a local‎ ‎ring $R$ where $\sigma$ is an endomorphism of $R$‎. ‎We show that‎ ‎$T_2(R,\sigma)$ is strongly clean if and only if for any $a\in‎ ‎1+J(R)‎, ‎b\in J(R)$‎, ‎$l_a-r_{\sigma(b)}‎: ‎R\to R$ is surjective‎. ‎Further‎, ‎$T_3(R,\sigma)$ is strongly clean if‎ ‎$l_{a}-r_{\sigma(b)}‎, ‎l_{a}-r_{\sigma^2(b)}$ and‎ ‎$l_{b}-r_{\sigma(a)}$ are surjective for any $a\in U(R),b\in‎ ‎J(R)$‎. ‎The necessary condition for $T_3(R,\sigma)$ to be strongly‎ ‎clean is also obtained‎.