The categories \(L\)-\(\mathsf{Frm}\) of lattice-valued frames (briefly, \(L\)-frames) were introduced by \textit{A. Pultr} and \textit{S. Rodabaugh} in [``Category theoretic aspects of chain-valued frames. I, II'', Fuzzy Sets Syst. 159, No.~5, 501--528 (2008; Zbl 1170.18004) and ibid. 529--558 (2008; Zbl 1170.18005)] for \(L\) a complete chain. They are an abstraction of the distinctive properties of the system of level mappings from an \(L\)-topology \(\tau\) into the topology \(\iota_L(\tau)\) given by the well-known iota functor \(\iota_L:L\text{-}\mathsf{Top}\rightarrow \mathsf{Top}\). A uniform enrichment of these categories was introduced by \textit{J. Gutiérrez García, I. Mardones-Pérez, M. A. de Prada Vicente} and the reviewer in [``Uniform-type structures on lattice-valued spaces and frames'', Fuzzy Sets Syst. 159, No.~19, 2469--2487 (2008; Zbl 1222.54032)]. In the paper under review, the authors address the problem of relaxing the condition of a complete chain raised by Pultr and Rodabaugh in the aforementioned articles. They start by formulating \(L\)-\(\mathsf{Frm}\) for more general underlying lattices \(L\), specifically completely distributive lattices, and then investigate whether or not this condition on lattices \(L\) can be further relaxed. They conclude that, among frames, this is not possible if the system of level mappings of any \(L\)-topology is required to be an \(L\)-frame (as it should).