Maximal divisibility-index conjecture for partial sums of Pell numbers

Let $P_n$) be the Pell numbers defined by $P_n=2P_{n-1}+P_{n-2}$ for $n>1$, with $P_0=0$ and $P_1=1$. Let $S^P_n$ denote the sum of the first $n$ Pell numbers, and let $m^P_n$ be the largest index $m$ such that $P_m$ divides $S^P_n$. Pell maximality conjecture. For the Pell sequence, we have

$$m^P_{4k+1}=m^P_{4k+2}=1,\quad m^P_{4k-1}=2k,\quad\text{and}\quad m^P_{4k}=2k+1.$$

The table of computed values exhibits this pattern, but no proof or resolution is supplied in the source.