Yuan's conjecture on simultaneous Pell equations with multiple solutions

Let $a,b,c,d$ be positive integer coefficients, and consider the system

$$a x^2-b y^2=1,\qquad c y^2-d z^2=1.$$

For positive integers $l$ and $m$, define

$$n(l,m)=\cfrac{\left(m+\sqrt{m^2-1}\right)^{2l}-\left(m-\sqrt{m^2-1}\right)^{2l}}{4\sqrt{m^2-1}},$$

and let $b(l,a)$ satisfy

$$4b(l,a)-1=\cfrac{\left(\sqrt a+\sqrt{a-1}\right)^l-\left(\sqrt a-\sqrt{a-1}\right)^l}{2\sqrt{a-1}},\qquad l\equiv3\pmod 4.$$

Yuan's conjecture. If this system has at least two solutions in positive integers, then its coefficients are given by

$$(a,b,c,d)=(1,m^2-1,1,n(l,m)^2-1),\qquad (a,a-1,b(l,a),b(l,a)-1),$$

or by equivalent forms.

The conjecture gives a classification of the exceptional simultaneous Pell systems that can have multiple positive-integer solutions; the paper presents it as a reformulation by Cipu and Mignotte of a conjecture of Yuan. The supplied text does not establish its resolution status.