= Solution
The bounded uniform <prior distributions> on the slopes and population intercept are proper and broad on their stated scale. They allow either sign but are not invariant to changes of units. The residual <precision parameter> $q=1/\sigma^2$ has a proper shape-rate $\operatorname{Gamma}(0.001,0.001)$ prior. Equivalently, $\sigma^2$ has <inverse-gamma distribution> with the same shape and scale. This is a commonly used diffuse choice, but it is not literally noninformative and its tail behaviour merits sensitivity analysis.
The prior for $\tau$ is uniform on $(0,100)$, so the induced density of $v=\tau^2$ is
$$
\boxed{\pi(v)=\frac1{200\sqrt v},\qquad0<v<10000.}
$$
It is integrable at zero, unlike the usual scale-invariant variance kernel $\pi(v)\propto1/v$. The latter kernel, often called the standard variance <Jeffreys prior>, would cause an <improper posterior from a log-uniform random-effect scale prior> here. After integrating the child intercepts, each child's response vector has a normal likelihood with <covariance matrix> $\sigma^2I_3+v\mathbf1\mathbf1^T$. For fixed positive $\sigma^2$, this covariance remains nonsingular at $v=0$, and the marginal likelihood has a positive finite limit there. On a compact set of the other parameters its positivity gives
$$
\int_0^\varepsilon L(v)\frac{dv}{v}=\infty.
$$
Consequently the scale-invariant kernel cannot be normalized to a <posterior distribution>. A proper prior on the standard deviation avoids this problem. This familiar scale prior is not necessarily the actual <Jeffreys prior for an additive variance component> derived from the observed-data likelihood; the latter can be finite at zero.
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