Infinite square-sequence Frobenius formula for large shifts

Let $A=(a,a+1^2,a+2^2,a+3^2,\ldots)$ be the infinite shifted square sequence. Let $\iota(n)$ denote the least number of squares of nonnegative integers whose sum is $n$.

Large-shift infinite square-sequence conjecture. If $a>30$, then

$$g(A)=3a+\max\{r\mid \iota(r)=4,\ \iota(a+r)\geq 3,\ \iota(2a+r)\geq 2\}.$$

The conjecture is motivated by computational examination of tables in the paper and is not presented there as proved. It concerns the Frobenius number of the infinite shifted square sequence and the role of the four-square representation function.