Skew circuits and circumference in a binary matroid

Let C_1 and C_2 be skew circuits in a binary matroid having circumference c. For any positive integer k there is a constant a_k such that if min { |A| ; C_1 ⊂ A ⊂ E-A} > a_k, then |C_1| + |C_2| < 2c -k.