Dissipativity conjecture for products of conservative interval maps without acips

Let $I$ be an interval, let $f:I\to I$ be an interval map, and let $\lambda$ be Lebesgue measure. Suppose that $(I,f,\lambda)$ is conservative and has no absolutely continuous invariant probability measure. Let $k_x$ be the induced time associated to the induced map in Theorem~, and suppose that $k_x$ is non-integrable with respect to Lebesgue measure and satisfies

$$\lambda(\{x:k_x>s\})\geq \frac{1}{\log s}.$$

Dissipativity conjecture. The product system $(I^2,f_2,\lambda_2)$ is dissipative.

The conjecture proposes conditions under which almost every pair may be Li--Yorke without having a dense orbit, by forcing the product measure to fail conservativity. The existence of conservative interval maps without acips and with exotic induced-time behavior was established in earlier work, but the supplied source does not establish this specific product dissipativity claim.