Additional quadruples (a, b, r, s) for totient-power representations

Determine whether there exist positive integer quadruples (a, b, r, s), other than (1, 1, 2, 2) and those satisfying gcd(ar, bs) = 1, such that every positive rational number can be written as ((φ(m^r))^a)/((φ(n^s))^b) for some m, n ∈ ℕ.

Background

Work on representing positive rational numbers via Euler’s totient function began with Krachun and Sun, who showed that every positive rational q can be written as φ(m^2)/φ(n^2). Li, Yuan, and Bai analyzed generalized forms φ(k m^r)/φ(l n^s) and gave complete criteria. Vu later considered the power form ((φ(m^r))^a)/((φ(n^s))^b), proving that if gcd(ar, bs) = 1, then every positive rational number admits such a representation.

The present paper provides new families of quadruples, notably (a, 1, 2, 2) and (1, b, 2, 2) with odd a, b > 1, for which every positive rational number can be expressed in the form ((φ(m^r))^a)/((φ(n^s))^b). However, a complete classification beyond the gcd(ar, bs) = 1 and (1,1,2,2) cases remains unresolved, motivating the explicit open problem stated.