Establish the conjectured \(t/2\)-coefficient outside the power-of-two obstruction

Prove or disprove that \(\mathrm{ex}^*(n,B_{t,3})=\frac{t}{2}n+O(1)\) whenever \(t+2\) is not a power of \(2\), and, at minimum, prove or disprove the corresponding assertion whenever \(t\not\equiv 2\pmod 4\).

Background

The paper establishes the $t/2$ asymptotic in several families, including odd $t$, powers of two, and selected values one less than powers of three, while also proving upper bounds for multiples of four. These results suggest that the exceptional behavior may be concentrated among parameters related to powers of two or the congruence class $2\pmod 4$. The authors explicitly formulate two conjectural extensions, while noting that suitable extremal constructions are the principal obstacle.