From an inaccessible cardinal, a strongly non-saturated Aronszajn tree can coexist with the non-existence of weak Kurepa trees.
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A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees
From an inaccessible cardinal, a strongly non-saturated Aronszajn tree can coexist with the non-existence of weak Kurepa trees.