Rational number or transcendental number, but not algebraic irrational number

Let P(n) and Q(n) be two non-trivial polynomials in n with rational coefficients and z[P, Q] is the value of infinite sum of P(n)/Q(n) from n=1 to +∞ (only when it converges, in which the degree of Q should be larger than or equal to the degree of P plus 2).

Claim: It is impossible for z[P,Q] to assume an algebraic irrational value (square root of 5, plastic ratio number, etc.). In other words, z[P, Q] is always either rational or transcendental. Some particular examples are:

P(n)=n, Q(n)=n³+2n² ⇒ z[P, Q] = 3/4 (rational)

P(n)=1, Q(n)=n⁴ ⇒ z[P, Q] = π⁴/90 (transcendental)

P(n)=16, Q(n)=n³+8n²+12n ⇒ z[P, Q] = 41/30 (rational)

P(n)=8, Q(n)=n²+4 ⇒ z[P, Q] = 2π*coth(2π) - 1 (transcendental)

Question: Is the above claim currently a proven fact (theorem) or just a conjecture/ hypothesis? If it is unproven, is there any progess in proving it recently?

Significance: If the above claim is true, one can show that z[P, Q] is transcendental by proving is irrationality. It will follows that:

• the Apéry constant ζ(3) is transcendental (by letting P(n)=1, Q(n)=n³)
• at least one of the numbers ζ(5), ζ(7), ζ(9) and ζ(11) is transcendental (Wadim Zudilin already proved that at least one of them is irrational in this interesting 2001 paper https://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng)
• Infinite numbers of the form ζ(2k+1) are transcendental (in which k is a positive integer and ζ denotes the Riemann zeta function)