= Solution
Let $M=\max\{b,y_1,\ldots,y_n\}$. The <continuous uniform distribution> likelihood contributes $\phi^{-n}\mathbf1_{\{\phi\geq\max_i y_i\}}$, while the <Pareto distribution> prior contributes $\phi^{-a-1}\mathbf1_{\{\phi\geq b\}}$. Their product proves <Uniform-Pareto conjugacy>:
$$
\boxed{p(\phi\mid\boldsymbol y)=(a+n)M^{a+n}\phi^{-(a+n+1)}\mathbf1_{\{\phi\geq M\}}}.
$$
Thus the <posterior distribution> is Pareto with shape $a+n$ and lower bound $M$. Its density is strictly decreasing on its support, so the <highest posterior density interval> begins at $M$. Since the upper-tail probability is $(M/u)^{a+n}$, the required upper endpoint satisfies $(M/u)^{a+n}=0.05$. The <Pareto highest density interval> is therefore
$$
\boxed{[M,\ M\,20^{1/(a+n)}]}.
$$
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