Homologically undetectable entropy for Kummer diffeomorphisms

Let $M$ be the K3 surface obtained in the Kummer construction, and let $f_T$ be the diffeomorphism associated with $T\in\operatorname{SL}(4,\mathbb{Z})$. Suppose that $T$ has exactly three eigenvalues $\lambda_1,\lambda_2,\lambda_3$ outside the unit circle, ordered so that

$$|\lambda_1|\geq |\lambda_2|\geq |\lambda_3|>1.$$

Homologically undetectable entropy. The minimal entropy of any $g\in\operatorname{Diff}(M)$ isotopic to $f_T$ satisfies

$$\operatorname{h_{\rm top}}(g)>\log\operatorname{specrad}[g_*:H_*(M;\mathbb{R})\to H_*(M;\mathbb{R})],$$

with the difference between the two sides being close to, or equal to, $\log|\lambda_3|$.

This would give a negative answer to Shub's question of whether the homological lower bound for topological entropy is always realized, or can at least be approached by diffeomorphisms in a fixed isotopy class. The proposed excess is associated with entropy not detected by the homological action on the K3 surface.