Odd-index period formula conjecture for the family $1(01)^k$

Consider the family of finite configurations whose current binary word is $1(01)^k$, with period function $P(k)$ assigning to the configuration indexed by $k$ its period.

Odd-index period formula conjecture. The period function satisfies

$$P(2i+1)=4^{2i+3}-1$$

for $i\in\mathbb{N}$. The formula is suggested by computational experiments, and the source explicitly says that it is suspected to hold for every $i\in\mathbb{N}$; no proof or resolution is given.