Variant of Ramanujan polynomials and a conjecture of Maji & Sarkar

We settle a conjecture proposed by B. Maji and T. Sarkar [ <italic>Zeros of Ramanujan-type polynomials</italic> , arXiv:2306.10283] regarding the location of zeros of a two-parameter family of reciprocal polynomials, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript k comma script l Baseline left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ℓ </mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_{k,\ell }(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for positive integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These polynomials are generalizations of Ramanujan polynomials studied by M. R. Murty, C. Smyth, and R. Wang [J. Ramanujan Math. Soc. 26 (2011), pp. 107–125]. More specifically, we show that except for two real zeros, all other zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript k comma script l Baseline left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi> ℓ </mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_{k,\ell }(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lie on the unit circle.