Divisibility patterns for terminal segments of Motzkin paths

Let $M=(\ldots,e_1,\ldots,e_k)$ be a Motzkin path of length $n+k$ whose last $k$ steps are $e_1,\ldots,e_k$.

Terminal-pattern divisibility conjecture. The following implications hold:

$$\begin{gathered} M=(\ldots,1,0,-1,-1)\Rightarrow (l+2n+5)\mid w_l(M),\\ M=(\ldots,1,0,0,-1,-1)\Rightarrow (l+2n+7)\mid w_l(M),\\ M=(\ldots,1,1,-1,-1,-1)\Rightarrow (l+2n+2)(l+2n+7)(l+2n+8)\mid w_l(M),\\ M=(\ldots,1,1,-1,0,-1,-1)\Rightarrow (l+2n+2)(l+2n+7)\mid w_l(M),\\ M=(\ldots,1,1,-1,0,0,-1,-1)\Rightarrow (l+2n+2)\mid w_l(M),\\ M=(\ldots,1,1,-1,0,0,0,-1,-1)\Rightarrow (l+2n+2)\mid w_l(M),\\ M=(\ldots,1,0,1,-1,-1,-1)\Rightarrow (l+2n+8)\mid w_l(M),\\ M=(\ldots,1,1,0,-1,-1,-1)\Rightarrow (l+2n+2)(l+2n+8)^2\mid w_l(M),\\ M=(\ldots,1,1,0,-1,0,-1,-1)\Rightarrow (l+2n+2)\mid w_l(M),\\ M=(\ldots,1,1,0,-1,0,0,-1,-1)\Rightarrow (l+2n+2)\mid w_l(M),\\ M=(\ldots,1,1,0,-1,0,0,0,-1,-1)\Rightarrow (l+2n+2)\mid w_l(M),\\ M=(\ldots,1,1,0,0,-1,-1,-1)\Rightarrow (l+2n+2)(l+2n+10)\mid w_l(M). \end{gathered}$$

These proposed divisibility relations connect rational roots of $w_l(M)$ with the ending of the Motzkin path. The supplied text gives no resolution.