## 1 Introduction  
Chronic insomnia and obstructive sleep apnoea (OSA) are the two most common sleep disorders and frequently co-occur [1, 2]. Chronic insomnia is characterised by self-reported difficulty initiating sleep, maintaining sleep, and/or early morning awakenings, combined with daytime impairments that persist for at least 3 months [1, 3, 4]. Chronic insomnia occursin approximately 6-15% of the general population [4; 5]. Obstructive sleep apnoea (OSA) is characterised by frequent narrowing (hypopnoea) and closure (apnoea) of the upper airway during sleep [6]. These events cause hypoxemia, cortical arousals and awakenings from sleep, and daytime impairments [6]. Moderate OSA occurs in approximately 6-17% of the general population [7]. Co-morbid insomnia and sleep apnoea (COMISA) is a prevalent and debilitating sleep disorder that is more difficult to treat compared to either insomnia or OSA alone [2; 8]. For example, approximately 30-40% of people with insomnia have co-morbid OSA and 30-50% of sleep clinic patients with OSA report insomnia symptoms [9; 10]. COMISA is associated with worse sleep, increased daytime impairments, poorer mental and physical health, and reduced longevity, compared to people with neither disorder or those with insomnia or OSA alone [8; 11; 12]. People with insomnia alone generally underestimate their sleep time, overestimate how long it takes to initially fall asleep, and overestimate the time they spend awake throughout the night, compared to polysomnography-derived estimates of these parameters [13]. This phenomenon is referred to as subjective-objective sleep discrepancy (SOSD, but also \"sleep misperception\", \"sleep underestimation\", and \"sleep-state discrepancy\"), and may contribute to the development and maintenance of insomnia [14; 15; 16; 17]. For example, [PERSON] and [PERSON] (1995) [18] and [PERSON] et al. (2019) [14] found that individuals with insomnia underestimated total sleep time by approximately 50 and 100 min, respectively [18]. Cognitive behavioural therapy for insomnia (CBT-I) is an effective and recommended \"first-line\" treatment for insomnia [19]. CBT-I is a multi-component therapy including behavioural, cognitive, and education components typically delivered over 4-8 weekly treatment sessions [19; 20; 21; 22]. Furthermore, CBT-I in patients with insomnia alone leads to large improvements in subjective sleep estimates and improvements in assessments of daytime functioning and feelings, despite little change in objective total sleep time. This results in a convergence of the two measures post-treatment, and a reduction in SOSD, meaning that subjective and objective measures align more closely [14; 23; 24; 25]. However, few studies have investigated SOSD in patients with COMISA and no study has investigated the effect of CBT-I on SOSD in a COMISA sample. SOSD may reflect bi-directional associations between insomnia and OSA [9]. For example, periods of brief sleep and awakenings following respiratory events may be (mis)perceived as periods of prolonged wakefulness [13]. Perceptions of long periods of wakefulness may increase frustration and cognitive arousal, contributing to an increased duration of light sleep [9]. In turn, increased time in light and transitional sleep may increase the likelihood of upper airway collapse [26; 27], and exacerbation of each condition [9]. [PERSON] et al. (2013) [28] investigated total sleep time SOSD in 92 people with insomnia alone, 66 with OSA alone, and 119 with COMISA and found the insomnia alone group significantly under-estimated total sleep time compared to objective recordings (67 \(\pm\) 87 min), but the OSA alone (17 \(\pm\) 78 min) and COMISA groups (36 \(\pm\) 88 min) did not [28]. Similarly, [PERSON] et al. (2016) [29] found that an insomnia alone group showed greater SOSD than an OSA alone group but not a COMISA group [29]. In a comparison of COMISA, insomnia alone, OSA alone, and no insomnia/OSA groups, [PERSON] et al.