A k-net of order n is an incidence structure consisting of n² points and nk distinguished subsets called lines, such that every line has exactly n points; parallelism is an equivalence relation on lines; k parallel classes exist, each consisting of n lines; and any two nonparallel lines meet exactly once. Moorhouse conjectures that if N _{k - 1} ⊂ N _k are (k−1)- and k-nets of order n, then rank _p N _k - rank _p N _{k - 1} ≥ n - k + 1 for any prime p dividing n at most once. The author establishes this conjecture for k = 3 using characters of loops and proves that if the conjecture is true, then any projective plane of order n ≡ 2 mod 4 or of squarefree order n is in fact desarguesian of prime order.