The finite test set conjecture for sums of triangular numbers representing odd integers

Let $f$colon be a sum of triangular numbers, and say that $f$ represents an integer $n$ when there is an admissible choice of variables for which $f=n$. The claim concerns representation of the set of all odd positive integers.

Finite test set conjecture. A sum of triangular numbers $f$ represents all odd integers if and only if it represents the integers

$$1,5,7,9,11,13,17,19,25,29,35,49,89.$$

This conjecture proposes an explicit finite test set for odd integers; the surrounding theorem establishes only the general existence of some finite subset, while the displayed set is supported by a computer calculation and its sufficiency remains to be proved.