We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{ [1]}(q^{ |R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{ i} in the relation { \cal R}_i = U_i{ \cal R}_1U_i^{ -1} between the i-th and the first generators { \cal R}_i of the braid group are universally expressible through the eigenvalues of { \cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
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