Conjectured rational points on genus-two curves associated with products of consecutive integers

Let $T\in A_5$, and let $C_T$ be the corresponding genus-two curve, with $r(T)$ denoting the rank quantity used in the source. Rational-points conjecture. For $r(T)>1$, the values of $T$ and the finite rational points $C_T(\mathbb{Q})$, omitting the points at infinity $(1,\pm1,0)$, are exactly those listed in the table in the source: $(2)$ with rank $2$ and points $\{(-2,0),(-1,0),(0,0),(2/3,\pm460/3)\}$; $(3)$ with rank $2$ and points $\{(-9,\pm252),(-3,0),(-2,0),(-1,0),(0,0),(-18/5,\pm468/5)\}$; $(0,4)$ with rank $4$ and points $\{(0,0),(1,\pm29),(9/4,\pm5871/4)\}$; $(1,3)$ with rank $2$ and points $\{(-4,\pm6),(-1,0),(0,0)\}$; $(1,4)$ with rank $2$ and points $\{(-1,0),(0,0)\}$; $(0,1,2)$ with rank $2$ and points $\{(-2,0),(0,0),(1,\pm27)\}$; $(0,2,3)$ with rank $2$ and points $\{(0,0)\}$; $(1,2,3)$ with rank $2$ and points $\{(-3,0),(-1,0),(0,0)\}$; $(1,3,4)$ with rank $2$ and points $\{(-4,\pm6),(-1,0),(0,0),(-25/9,\pm620/9)\}$; $(0,1,2,3)$ with rank $3$ and points $\{(-4,0),(-2,0),(-1,0),(0,0),(-13/3,\pm91/3),(-5/3,\pm55/3)\}$; and $(0,1,3,4)$ with rank $2$ and points $\{(-3,0),(-1,0),(0,0)\}$. The table is based on numerical computations after the authors were unable to characterize the rational points for some $T\in A_5$; it remains conjectural.