Conjecture on multiple roots of sums of products of consecutive integers

Let $nin\mathbb{N}$ and let $T=(a_{1},\ldots,a_{k})\in A_{n}$, where $A_n$ is the set of strictly increasing tuples with entries less than $n$. Define

$$g_T(x)=p_n(x)+\sum_{i=1}^k p_{a_i}(x),\qquad p_a(x)=\prod_{i=0}^{a}(x+i).$$

Multiple-root conjecture. The polynomial $g_T(x)$ has multiple roots if and only if one of the following holds: (1) $T=(n-4)$ for $n\geq4$, with $(x^2+(2n-3)x+n^2-3n+1)^2\mid g_T(x)$; (2) $T=(n-3,n-2)$ for $n\geq3$, with $(x+n-1)^3\mid g_T(x)$; or (3) $T=(n-2,n-1)$ for $n\geq2$, with $(x+n)^2\mid g_T(x)$. In each case, the corresponding cofactor has no multiple roots. The conjecture is motivated by computations and is intended to characterize the multiple-root configurations governing finiteness results for the associated Diophantine equations; no proof or resolution is supplied.