Baruchel's conjecture on the zeroth digit in base-phi representations

Let $N$ be a natural number with unique base-$\varphi$ representation, where $\varphi=(1+\sqrt{5})/2$ and the digits $d_i(N)$ are either $0$ or $1$, with no two consecutive digits equal to $1$. The notation $\lfloor\cdot\rfloor$ denotes the floor function.

Baruchel's conjecture. The digit $d_0(N)=1$ if and only if

$$N=\lfloor n\varphi\rfloor+2n+1$$

for some natural number $n$, or $N=1$.

This conjecture gives a precise description of the natural numbers whose zeroth base-phi digit is $1$, refining the known frequency result for this digit. The paper states that it is proved in Section 2.