Louboutin's polynomial identity conjecture for S and R

For $d\geq 1$, define

$$P_d(X,Y)=d\sum_{\substack{k,l\geq 0\\0\leq 2k+3l\leq d}}(-1)^{k-1}\binom{k+l}{k}\binom{d-k-2l}{k+l}\frac{X^kY^{d-2k-3l}}{d-k-2l}.$$

Let $a,b\in\mathbb{Z}$ be nonzero, put $c:=a+b\neq 0$, and define

$$S_{a,b}(T)=T^{-a}+T^{-b}+T^{a+b}.$$

For $d\in\{a,b,c\}$,

$$P_{|d|}(S_{a,b}(T),S_{a,b}(1/T))=-S_{a,b}(1/T^{|d|}).$$

If $a$ is even and $b$ is odd, also define

$$R_{a,b}(T)=T^{-a}-(-1)^{a+b}T^{-b}+T^{a+b}.$$

Then

$$P_{|d|}(-R_{a,b}(T),-R_{a,b}(1/T))=\begin{cases}-S_{a,b}(1/T^{|d|}),&d=a,\R_{a,b}(1/T^{|d|}),&d\in\{b,c\}.\end{cases}$$

This identity is one of Louboutin's conjectures used in the proof of the paper's main theorem; its resolution status is not specified in the supplied text.