The test is explained and described in the forum of the GIMPS project: https://www.mersenneforum.org/showthread.php?t=28658
In short: $$x_1=W_3=3 \ , \ x_{j+1} = \frac{x_j^4+2x_j^2+1}{4(x_j^3-x_j)}$$
$$\text{If } \ x_{q-1} \ \equiv \ -1/3 \pmod{W_q} \ , \ \text{then } W_q = \frac{2^q+1}{3} \text{ is PRobably Prime} .$$
This candidate PRP test for Wagstaff numbers ($(2^q+1)/3$ with $q$ prime) is inspired by the following primality test for Fermat numbers proved by Denomme-Savin and Tsumura by means of the Elliptic Curves theory:
$$x_1=F_1=5 \ , \ x_{j+1} = \frac{x_j^4+2x_j^2+1}{4(x_j^3-x_j)}$$
$$\text{If } \ x_{2^{n-1}} \equiv -1 \pmod{F_n} \ , \ \text{then } F_n = 2^{2^n}+1 \text{ is prime} .$$
See Posts #1, #4 and #5 for all the details.
See Post #1 for the context.
See Post #4 for the final version of the test.
See Post #5 for the final version of the Pari/gp code.