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Dorsal closure is a process that occurs during embryogenesis of Drosophila melanogaster. During dorsal closure, the amnioserosa (AS), a one-cell thick epithelial tissue that fills the dorsal opening, shrinks as the lateral epidermis sheets converge and eventually merge. During this process, the aspect ratio of amnioserosa cells increases markedly. The standard 2-dimensional vertex model, which successfully describes tissue sheet mechanics in multiple contexts, would in this case predict that the tissue should fluidize via cell neighbor changes. Surprisingly, however, the amnioserosa remains an elastic solid with no such events. We here present a minimal extension to the vertex model that explains how the amnioserosa can achieve this unexpected behavior. We show that continuous shrinkage of the preferred cell perimeter and cell perimeter polydispersity lead to the retention of the solid state of the amnioserosa. Our model accurately captures measured cell shape and orientation changes and predicts non-monotonic junction tension that we confirm with laser ablation experiments.