A sample $(n_1, n_2,\cdots, n_t)$ is taken from a $t$-category multinomial population. The hypothesis of equiprobability, that the $t$ physical probabilities associated with the cells are all equal to $1/t$, is called the null hypothesis. Conditional on the non-null hypothesis, a symmetric Dirichlet prior of parameter $k$ is assumed $(k \geqq 0)$ and the Bayes factor against the null hypothesis, with this assumption, is denoted by $F(k)$. A conjecture made in 1965 is almost proved, namely that $F(k)$ has a unique local maximum and that this occurs for a finite value of $k$ if and only if Pearson's $X^2$ exceeds its number of degrees of freedom. The result is required for the calculation of $\max F(k)$, which provides a non-Bayesian significance criterion whose simple asymptotic distribution is good even in the extreme tail, and even for sample sizes less than $t$. This criterion arose from an attitude involving a Bayes/non-Bayes compromise.