Is the Shapiro inequality for $n=23$ an open problem? The reason why I am asking is I have two contradictory pieces of information from two different articles.

The first article titled "The validity of Shapiro’s cyclic inequality" published by B.A. Troesch in the journal "Mathematics of Computation" in 1989 claims that the inequality does indeed holds for all odd $N \leq 23$. The link to the article is here.

However, there is another article published in 2002, titled "Shapiro's Cyclic Inequality for Even n" by Bushell, PJ and McLEOD, JB in the journal "Journal of Inequalities and Applications" that claims that the case $n=23$ is still an open problem. Here is a link to the article. Interestingly, this article cites the previous article.

The Shapiro inequality is the claim that if $x_1,x_2,\ldots,x_n \in \mathbb{R}^+$, then $$\sum_{k=1}^{n} \dfrac{x_k}{x_{k+1} + x_{k+2}} \geq \dfrac{n}2$$ where $x_{n+1} = x_1$ and $x_{n+2} = x_2$ holds true for $n \in \{1,2,3,\ldots,12,13,15,17,19,21,23 \} $.

Here is the wikipedia link for whatever it is worth.