Polynomial-scale density conjecture for ideal degree-three Prouhet–Tarry–Escott solutions

Let $N(H)$ denote the number of nontrivial ideal degree-three Prouhet–Tarry–Escott solutions of size four and height at most $H$, counted up to permutations within each multiset and interchange of the two multisets, with distinct integer translates counted separately.

Polynomial-scale density conjecture. The full counting function satisfies

$$N(H)=H^{3+o(1)}.$$

Consequently, since the symmetric solutions satisfy

$$N_{\mathrm{sym}}(H)=\frac{4\log 2}{3\pi^2}H^3\log H+O(H^3),$$

the symmetric locus accounts for the full solution space up to subpolynomial factors:

$$\frac{N_{\mathrm{sym}}(H)}{N(H)}\geq H^{-o(1)}.$$

The conjecture predicts that the symmetric locus has the same height exponent as the full solution space, although its literal proportion may tend to zero.