Summary: For normal models with \(X\sim N_p(\theta,\sigma^2 I_p)\), \(S^2\sim\sigma^2\chi^2_k\), independent, we consider the problem of estimating \(\theta\) under scale invariant squared error loss \(\| d-\theta\|^2/\sigma^2\), when it is known that the signal-to-noise ratio \(\|\theta\|/\sigma\) is bounded above by \(m\). Risk analysis is achieved by making use of a conditional risk decomposition and we obtain in particular sufficient conditions for an estimator to dominate either the unbiased estimator \(\delta_{UB}(X)=X\), or the maximum likelihood estimator \(\delta_{ML}(X,S^2)\), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator \(\delta_{BU,0}\) associated with a prior on \((\theta,\sigma^2)\) such that \(\theta|\sigma^2\) is uniformly distributed on the (boundary) sphere of radius \(m\sigma\) and a non-informative \(1/\sigma^2\) prior measure is placed marginally on \(\sigma^2\). With a series of technical results related to \(\delta_{BU,0}\); which relate to particular ratios of confluent hypergeometric functions; we show that, whenever \(m\leq\sqrt{p}\) and \(p\geq 2\), \(\delta_{BU,0}\) dominates both \(\delta_{UB}\) and \(\delta_{ML}\). The finding can be viewed as both a multivariate extension of \(p=1\) result due to [T. Kubokawa, Sankhyā 67, No. 3, 499–525 (2005; Zbl 1192.62062)] and an unknown variance extension of a similar dominance finding due to [É. Marchand and F. Perron, Ann. Stat. 29, No. 4, 1078–1093 (2001; Zbl 1041.62016)]. Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for \(m\leq\sqrt{p/2}\), a wide class of Bayes estimators, which include priors where \(\theta|\sigma^2\) is uniformly distributed on the ball of radius \(m\sigma\) centered at the origin, are shown to dominate \(\delta_{UB}\).