The Answer is Yes. In order for the Secretary of State to be able to rely on evidence of misconduct in collateral companies to try and secure a disqualification, it must firstly show some form of misconduct in respect of the lead company.
The Secretary of State cannot simply disqualify a director solely on the basis of conduct in collateral companies if there is no evidence of misconduct in the lead company.
However, it is important to understand that in respect of the lead company, the Secretary of State only needs to show there is some form of misconduct even if that misconduct does not constitute unfitness. The very fact there is some form of misconduct then enables the Secretary of State to introduce evidence of misconduct in the collateral companies, which if shown to constitute unfitness, can then form the basis of a disqualification.
If unfitness is actually established in respect of the lead company, then any additional unfit misconduct proved in the collateral companies as well will simply be used to increase the length of the disqualification imposed against the individual concerned.