Conjecture on sums of a practical number and two polygonal numbers

For $s\in\{4,5,6,7,8,10\}$, an $s$-gonal number is a number in the corresponding polygonal-number sequence, and a practical number is a natural number whose positive integers up to it can be represented as sums of distinct divisors. A sum of two $s$-gonal numbers allows the two summands to be chosen independently.

Conjecture on two polygonal summands. For $s\in\{4,5,6,7,8,10\}$, all natural numbers can be written as a sum of a practical number and two $s$-gonal numbers.

The conjecture is motivated by computational verification for all natural numbers below $10^8$ and is described in the source as a stronger version of a cited theorem for these values of $s$. The supplied status evidence says that the conjecture does not hold for other values of $s$; no counterexample to the stated finite set of values is supplied.