$n=x^2+y^2+z^2+w^2$ with $9x^2+16y^2+24z^2+48w^2$ a square

Lagrange’s four-square theorem states that each $ n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $ x^2+y^2+z^2+w^2$ with $ x,y,z,w\in\mathbb Z$ .

Question. Can we write each $ n\in\mathbb N$ with $ n\not=71,\ 85$ as $ x^2+y^2+z^2+w^2$ $ (x,y,z,w\in\mathbb N$ ) with $ 9x^2+16y^2+24z^2+48w^2$ a square?

In Jan. 2017 I raised this question (cf. http://oeis.org/A281659) and conjectured that it has a positive answer. For example, $ $ 11 = 3^2 + 1^2 + 1^2 + 0^2$ $ with $ $ 9\times3^2 + 16\times1^2 + 24\times1^2 + 48\times0^2 = 11^2,$ $ and $ $ 170 = 3^2 + 6^2 + 2^2 + 11^2$ $ with $ $ 9\times3^2 + 16\times6^2 + 24\times2^2 + 48\times11^2 = 81^2. $ $

It seems that all known proofs of Lagrange’s four-squre theorems cannot be adapted to yield a positive answer to the question. Your comments are welcome!