## $12n+5=x^2+y^2+z^2$ with $x\in\{2^a5^b:\ a=1,2,3,\ldots\ \text{and}\ b=0,1,2,\ldots\}$

By the Gauss-Legendre theorem on sums of three squares, for each $$m\in\mathbb N=\{0,1,2,\ldots\}$$, we may write $$4m+1$$ as the sum of three squares. Surprisingly, I found that this classical result seems too weak in the case $$m\equiv1\pmod3$$. Namely, on June 15, 2019 I formulated the following conjecture (cf. http://oeis.org/A308661).

Conjecture. Let $$n\in\mathbb N$$. Then we can write $$12n+5$$ as $$(2^a5^b)^2+c^2+d^2$$, where $$a,b,c,d\in\mathbb N$$ and $$a>0$$. Equivalently, $$3n+1$$ can be written as $$T_a+T_b+(2^c5^d)^2$$ with $$a,b,c,d$$ nonnegative integers, where $$T_k$$ denotes the triangular number $$k(k+1)/2$$.

I verified this for all $$n=0,\ldots,2\times10^8$$, and later Giovanni Resta extended the verfication for $$n<8.33\times10^9$$. For example, $$12\times441019 + 5 = 5292233 = (2^15^2)^2 + 513^2 + 2242^2 = (2^35^1)^2 + 757^2 + 2172^2.$$ It seems that we could not solve the conjecture via the theory of ternary quadratic forms.

Question. What tools might be helpful towards a proof of the conjecture?

Your further check of the conjecture is also welcome!