Let L denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G, and possibly also that of L, L may admit differentiable Lp-functional calculi, or may be of holomorphic Lp-type for a given p≠ 2. ‘Holomorphic Lp-type’ means that every Lp-spectral multiplier for L is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the L2-spectrum of L. This can in fact only arise if the group algebra L1(G) is non-symmetric. Assume that p≠ 2. For a point ℓ in the dual g* of the Lie algebra g of G, denote by Ω(ℓ)=Ad*(G)ℓ the corresponding coadjoint orbit. It is proved that every sub-Laplacian on G is of holomorphic Lp-type, provided that there exists a point ℓ∈ g* satisfying Boidol's condition (which is equivalent to the non-symmetry of L1(G)), such that the restriction of Ω(ℓ) to the nilradical of g is closed. This work improves on results in previous work by Christ and Müller and Ludwig and Müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group G, and on the other hand, for the case p>1, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits. It seems likely that the condition that the restriction of Ω(ℓ) to the nilradical of g is closed could be replaced by the weaker condition that the orbit Ω(ℓ) itself is closed. This would then prove one implication of a conjecture by Ludwig and Müller, according to which there exists a sub-Laplacian of holomorphic L1 (or, more generally, Lp) type on G if and only if there exists a point ℓ∈ g* whose orbit is closed and which satisfies Boidol's condition.