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Given a separable fibration over an algebraically closed field of positivecharacteristic, we study an Iitaka-type inequality for the anticanonicaldivisors. We conclude that it holds when the source of the fibration is athreefold or when the target is a curve. We then give counterexamples incharacteristics $2$ and $3$ for fibrations with "very singular" fibres,constructed from Tango-Raynaud surfaces.

Iitaka conjecture for anticanonical divisors in positive characteristics