The conjecture on the ternary sum of triangular numbers Δ₁,₄,₅

For positive integers $a_1,\a_2,\a_3$, write $\Delta_{a_1,a_2,a_3}$ for the corresponding sum of triangular numbers; in particular, $\Delta_{1,4,5}$ is the ternary sum with coefficients $1,4,5$. Conjecture on $\Delta_{1,4,5}$. The sum $\Delta_{1,4,5}$ represents every nonnegative integer except $2$. The conjecture is supported by computation through $10^7$, and Kane proved representation of all positive odd integers under the generalized Riemann hypothesis for weight-$2$ newforms; the full assertion remains open.