Positivity conjecture for the approximation exponents $f_t$

For integers $t\geq 3$, define

$$f_t = \inf_{b>a \geq 10} \frac{\log \min_{n \in \mathbb{N}} \left| \frac{a^t+b^t}{2}-n^t \right|}{\log |b^t-a^t|}.$$

The numerical evidence suggests that $f_3=0$ and $f_t>0$ for $t\geq 4$. Positivity conjecture. For integers $t\geq 4$, $f_t>0$. On the other hand, $f_3=0$ and there are infinitely many integer solutions to

$$x^3+y^3-2z^3 \in \{ \pm 1,\pm 2\}.$$

This conjecture is motivated by Mazur's near miss problem and concerns how closely perfect $t$-th powers can approximate arithmetic progressions. The source provides numerical evidence but no proof of the asserted positivity for $t\geq 4$.