## 1 Introduction  
Despite the global effort attempting to reverse the impacts associated with climate change a recent study by the United Nations shows that causes and impacts of such action are accelerating rather than slowing down [1]. The greenhouse gas emissions, in which carbon dioxide plays a key role, have increased nearly 20% in the last 5 years (2015-2019) compared with the period 2011-2015. The CO\({}_{2}\) emissions from the combustion of fossils fuels and those arising from cement production hugely contribute to the growth of the anthropogenic carbon. The persistence of existing strategies will have devastating consequences for the environment, and therefore urgent actions to tackle climate change are imperative. The building sector might play a vital role in climate change mitigation since it is responsible for a large share of the energy consumption (-40%) and CO\({}_{2}\) emissions (36%) in the European Union (EU) [2]. The energy efficiency of buildings deserves special attention considering that 75% of the EU buildings stock is energy inefficient [3]. To enhance the energy efficiency and decarbonize the building stock, new high-performance materials must be developed and used in building construction/rehabilitation. The use of lightweight concrete with low thermal conductivity is a promising approach to mitigate the energy losses inside buildings [4; 5]. Portland cement has been the most widely used ingredient in mortars and concretes over the past century. However, concerns regarding the CO\({}_{2}\) emissions associated with its production [6] have led to the pursuit of alternative binders. Inorganic polymers emerge as the most promising avenue having much lower embodied CO\({}_{2}\), provided that proper mixture design is employed [7; 8], and due to the possibility of using wastes as raw materials instead of commercial precursors [9] which further decreases the carbon footprint of these materials and alleviates the pressure exerted by the unprecedented volume of waste production estimated to reach a stunning 3.4 Bt in 2050 [10]. The feasibility of using lightweight inorganic polymers in low thermal-conductivity applications has been recently highlighted in review papers [5; 11]. Amongst the various synthesis protocols that might be employed to produce lightweight inorganic polymers, the chemical foaming technique [12; 13] and the use of lightweight aggregates [14; 15] are the most common routes. In the former a porous body is produced due to the decomposition of the foaming agent in the alkaline medium releasing gas bubbles which are then trapped in the paste, while in the latter a reduction of density depending on the nature and amount of the aggregate is seen in the specimens. The use of natural and sustainable lightweight aggregates instead of fossil fuel-derived ones is a particularly attractive way to enhance the global sustainability of the construction sector [16; 17]. Nevertheless, this possibility has hardly been considered and most of the studies used polymeric-based aggregates (e.g., polystyrene) [18; 19] made from fossil fuels which deviates from the sustainability paradigm envisioned for the construction of green buildings. Nonetheless, it should be highlighted that some studies have used polymer wastes as light aggregates (e.g., polyurethane wastes [20], crumb rubber from recycled tires [21]) partially diluting sustainability concerns. In any case, the use of natural and renewable aggregates is expected to provide environmental benefits over synthetic or non-renewable materials. Cork is a highly sustainable and natural material with a unique array of properties that set it apart from other materials. The sustainability of cork derives from the non-destructive harvesting from the oak tree (i.e., the bark is simply stripped from the tree) which has a lifespan of more than 200 years, and to the increase in the CO\({}_{2}\) uptake by the tree in the regeneration of a new cork layer.