tabctl32.msm installed by WiX does not register the ocx

I'm moving an old VB6 app from InstallShield to WiX installer driven by a Visual Studio WiX project. It's all good except that I need the Microsoft tab control from tabctl32.msm. When I include this in my WiX project, the OCX installs but does not work until I run regsvr32 on it.

Using either InstallShield or the Visual Studio Installer, the msm correctly registers its component. I assume that it's the msm's job to do that.

I am installing per-machine and the msi runs as administrator. Here's some simplified WiX code that illustrates the problem:

<?xml version="1.0" encoding="UTF-8"?>
<Wix xmlns="http://schemas.microsoft.com/wix/2006/wi">
    <Product Id="*" Name="TabCtlTest" Language="1033" Version="1.0.0.0" Manufacturer="A Company" UpgradeCode="bd70911b-976f-462d-b381-26bc4dc0758e">
        <Package InstallerVersion="200" Compressed="yes" InstallScope="perMachine" />

        <MajorUpgrade DowngradeErrorMessage="A newer version of [ProductName] is already installed." />
        <MediaTemplate EmbedCab="yes" />

        <Feature Id="ProductFeature" Title="TabCtlTest" Level="1">
            <ComponentGroupRef Id="ProductComponents" />
            <MergeRef Id="TABCTL32"/>
        </Feature>
    </Product>

    <Fragment>
        <Directory Id="TARGETDIR" Name="SourceDir">
            <Merge Id="TABCTL32" SourceFile="$(env.ProgramFiles)Common FilesMerge ModulesTABCTL32.MSM" Language="1033" DiskId="1" />
            <Directory Id="ProgramFilesFolder">
                <Directory Id="INSTALLFOLDER" Name="TabCtlTest" />
            </Directory>
        </Directory>
    </Fragment>

    <Fragment>
        <ComponentGroup Id="ProductComponents" Directory="INSTALLFOLDER">
            <Component Id="cmp59A149A53D5CFB2D09E03E213D57D2E4" Guid="{BA66D5C5-0813-4822-9F40-8E5C9B242E8B}">
                <File Id="fil36DDAE6925146DA614B90C0E87BD68AC" KeyPath="yes" Source="..vbprogProject1.exe" />
            </Component>
        </ComponentGroup>
    </Fragment>
</Wix>

I'm using Visual Studio 2015 and WiX 3.9.