Conjecture on the complexity of tromino tilings of extended domino-deficient rectangles

Let $N_T(m,n)$ denote the number of tromino tilings of an $m\times n$ rectangle. Consider an extended domino-deficient rectangle with minimal dimensions $m\times n'$, where $n'>n$. Complexity conjecture. The complexity of the number of tromino tilings changes from $N_T(m,n)$ to

$$\mathcal{O}((m+n)\cdot N_T(m,n)).$$

This conjecture proposes that introducing a minimal extension and a domino deficiency increases the tiling-count complexity by a factor linear in the sum of the rectangle dimensions. The preceding $4\times n$ analysis motivates the claim, but no general proof or resolution is given here.