Typically, setting your conversion window as wide as possible increases your chances of attributing a tracked conversion. LinkedIn recommend 90-day click, 90-day view for lead or lower-funnel conversions and 30-day click, 7-day view for website conversions.
Associate your Conversions API conversions to as many applicable campaigns as possible to maximize chances of attributing a conversion.
Consider using the Last Touch - Each Campaign model, which gives credit to all campaigns that had an ad interaction within that window. With this model, each conversion event counts across all campaigns that served an impression within the conversion window to the converting member.