Tail exponent conjecture for NESS under power-law resetting

Prove that for a one-dimensional Brownian particle subject to the position-dependent resetting rate r(x)=r0|x|^λ, the non-equilibrium steady-state density exhibits asymptotic decay of the form exp(−|x|^{λ/2+1}).

Background

By estimating NESS numerically for several values of λ, the authors observe that the tails fit the form e^{−|x|^α} with α depending on λ. From these observations, they propose the hypothesis α=λ/2+1 for the tail exponent.

They explicitly state that this hypothesis remains to be proven, framing a clear conjecture about the asymptotic behavior of NESS under power-law resetting that would unify and extend known special cases (λ=0 and λ=2).