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The procedure that was used in an earlier paper on the generalized Ces\`{a}romatrices of order two is adapted here to show that generalized Ces\`{a}romatrices of order three are supraposinormal on $\ell^2$. This leads toinformation about their posinormality, coposinormality, and hyponormality. Thereader is then invited to formulate a conjecture regarding generalizedCes\`{a}ro matrices of other orders.

The generalized Ces\`{a}ro matrices of order three are supraposinormal on $\ell^2$