I also understand that per CPR 38.5(3), withdrawing a Claim under Part 38 procedure does not affect proceedings to deal with any question of costs.
Does the costs provision under CPR 38.6(1) apply? I understand the below from access to my legal research database/application.
38.6 Liability for costs.
(1) Unless the court orders otherwise, a claimant who discontinues1 is liable for the costs which a defendant against whom the claimant discontinues incurred on or before the date on which notice of discontinuance was served on the defendant.
1 The court has to take into account the factors set out in CPR 44.2. A costs order will be deemed to have been made for costs to be assessed on the standard basis unless a party makes an application to reverse or vary the general rule: CPR 44.9. The defendant may wish to make application to override the automatic provisions of CPR 44.9 to seek an order for costs on the indemnity basis. Guidance on the issue of costs where there is discontinuance is comprehensively provided in Brookes v HSBC Bank plc [2011] EWCA Civ 354, [2012] 3 Costs LR 285 at [6]–[8] per Moore-Bick LJ.
The court should not, without good reason, depart from the usual rule that the discontinuing party should pay the costs of the other party and, if it does, it should state why: Walker v Walker [2005] EWCA Civ 247, [2006] 1 All ER 272, [2006] 1 WLR 2194. The impact of the Coronavirus pandemic as a reason for discontinuance is not, by itself, a sufficient reason for departing from the normal rule that the claimant should pay the defendant’s costs: Khan v Governor of HMP The Mount [2020] EWHC 1367 (Admin), [2020] All ER (D) 60 (Jul).
The fact that the discontinuing claimant would have, or might well have, succeeded at trial is not itself a good reason for disapplying the presumption that the defendant’s costs should be paid: Teasdale v HSBC Bank plc [2010] EWHC 612 (QB), [2010] 4 All ER 630, [2010] NLJR 878.
In general, a claimant who discontinues will have to bear the costs of the defendant’s claim to be indemnified by a third party. However, the court may order otherwise where the defendant’s claim is shown to have been unlikely to succeed: Young v J R Smart (Builders) Ltd (7 February 2000, unreported) (Trans Ref QBENI 99/0742/1), [2000] CLY 456, CA.
When exercising discretion, a court will consider all the circumstances but a defendant’s conduct is not unreasonable if during proceedings it exercises a contractual right to engage in mediation in relation to an aspect of the claim, and the fact of doing so does not amount to a change of circumstances such that the normal costs provisions applicable to a claimant’s discontinuance of that aspect of the claim should be disapplied: Epoq Legal Ltd v DAS Legal Expenses Insurance Co Ltd [2022] EWHC 1577 (Comm), [2022] Costs LR 1123.