Natural-extension conjecture for simple alternating $N$-expansions

Let $(T,\Omega,mu_T,\mathcal{B}_\Omega)$ be a simple system that is ergodic with $mu_T$ an absolutely continuous invariant measure, and let $\mathcal{B}_\Omega$ be the Borel $\sigma$-algebra on $\Omega$. Define

$$\mathcal{T}_T(x,y)=\left(T(x),\frac{N}{d_1(x)+y}\right)$$

on $\Omega\times[0,\infty)$, and set

$$X=\bigcap_{i=0}^{\infty}\mathcal{T}_T^i(\Omega\times[0,\infty)).$$

Natural-extension conjecture. The restriction $\overline{\mathcal{T}_T}:=\mathcal{T}_T|_X$ is bijective Lebesgue almost everywhere, and $(\overline{\mathcal{T}_T},X,\overline{\mu_T},\mathcal{B}_X)$ is the natural extension of $(T,\Omega,\mu_T,\mathcal{B}_\Omega)$, where $\overline{\mu_T}$ is an invariant measure absolutely continuous with respect to two-dimensional Lebesgue measure and satisfies

$$\overline{\mu_T}(A)=C\iint\limits_A\frac{N}{(N+xy)^2}\,dy\,dx$$

for $A\in\mathcal{B}_X$, with $\mathcal{B}_X$ the Borel $\sigma$-algebra restricted to $X$ and $C$ a normalising constant. The conjecture proposes a natural extension and an explicit invariant density for the simple case on more than two intervals; the source reports numerical support, but does not establish the claim in general.