We report an empirical conjecture on the asymptotic structure of degree- (2, 1) polynomial continued fractions (PCFs) with integer coefficients. Complete enumeration of F (2, 4) reveals 24 Trans-stratum families (limits that are transcendental and non-logarithmic), all satisfying the ratio identity a₂/b₁² = -2/9. Extended search over F (2, 5) confirms 56 Trans and 12 Log families, all with the same ratio. A targeted falsification test at denominator values b₁ in 4, 5 -- testing candidate ratios -3/16, -4/25, -6/25 arising from rational indicial analysis -- finds zero Trans or Log families across 7, 174 convergent candidates, bringing the empirical base to approximately 150, 000 families with zero counterexamples. We conjecture (Transcendental Ratio Identity) that for any convergent integer-coefficient degree- (2, 1) PCF whose limit is Trans-stratum, a₂/b₁² = -2/9, equivalently the associated three-term recurrence has indicial exponents 1/3, 2/3 at infinity. A separate Brouncker boundary class at ratio +1/4 (indicial double root 1/2, 14 families at D=5) is identified as a distinct stratum. The conjecture connects to Worpitzky/Thron-Waadeland limit-periodic CF theory and speculatively to Schwarz's list of algebraic hypergeometric functions. All computational data and scripts are available at the linked repository.
papanokechi (Mon,) studied this question.