Conjecture on sums of practical and polygonal numbers

For a natural number $s>3$, 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 natural number is sufficiently large if all natural numbers beyond some threshold have the stated property.

Conjecture on practical and polygonal sums. If $s>3$, $s \not\equiv 0\pmod{12}$, and $s \not\equiv 4\pmod{12}$, then all sufficiently large natural numbers can be written as a sum of a practical number and an $s$-gonal number.

The claim is based on computations of representations by a practical number and an $s$-gonal number. The source explicitly notes that it does not hold for other values of $s$, so the conjecture as stated is refuted.