$12n+5=x^2+y^2+z^2$ 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!