Strong transience criterion for the conditioned right step

Let $P$ be a distribution on environments satisfying Assumption asmiid, and suppose that

$$E_P[\log \rho_0] \in (-\infty,0).$$

Here, $\mathfrak{R}$ denotes the first return time to the origin, and $X_1$ is the first step of the random walk. Strong transience criterion. Under these assumptions,

$$\mathbb{E}[\mathfrak{R}\mid X_1=1,\,\mathfrak{R}<\infty]<\infty \quad\iff\quad E_P[\rho_0]<1.$$

The theorem preceding this conjecture proves the forward implication when $E_P[\rho_0]<1$ and proves the converse under the additional condition that either $E_P[\rho_0]>1$ or $E_P[\rho_0]=1$ with $E_P[\rho_0\log\rho_0]<\infty$. The conjecture asks whether the same equivalence holds under the weaker assumption $E_P[\rho_0]\geq 1$ without that additional integrability condition.