The sub-four-thirds exponent conjecture for Frobenius numbers

Let $(a,b,c)$ be an admissible triplet, meaning a triplet of positive integers satisfying the admissibility conditions for the three-variable Frobenius problem, and let $f(a,b,c)$ denote its Frobenius number. Sub-four-thirds exponent conjecture. There exists an upper bound for $(a,b,c)$ proportional to

$$\sqrt{abc}^{p}$$

where $p<\frac{4}{3}$, valid for all admissible triplets $(a,b,c)$. This conjecture seeks an upper bound of smaller magnitude than the known bounds, which are comparable to one proportional to $\sqrt{abc}^{4/3}$; the empirical evidence in the paper suggests the stronger exponent $p=\frac{5}{4}$ may work, but the general assertion remains unproved.