The binary-operation identity for the discrete polynomials S and T

Let $S$ and $T$ be the sets of discrete polynomials introduced above, and let the binary operation $\langle S,T\rangle^r_s(i)$ be defined by

$$\langle S,T\rangle^r_s(i)=\sum_{j=0}^{r}(-1)^jS^{r-j}_{s+j}(i)T^j_{s-r+j}(i+r-j).$$

The binary-operation identity. The relations

$$\langle S,T\rangle^r_s(i)=\sum_{j=0}^{r-2}(-1)^jT(i+r-j-2)T(i+s+j)S^{r-j-2}_{s+j+2}(i)T^j_{s-r+j}(i+r-j),$$

for all $s\geq r-1$, are identities. This identity is presented as an important conjecture for the binary operation on sets of discrete polynomials; the supplied text gives no resolution, so its status remains open.